Modified defect relation for Gauss maps of minimal surfaces with hypersurfaces of projective varieties in subgeneral position

TL;DR

This paper develops modified defect relations for the Gauss map of complete minimal surfaces in projective varieties, providing bounds on the number of hypersurfaces intersected, extending previous results in the field.

Contribution

It introduces new defect relations for Gauss maps of minimal surfaces with hypersurfaces in subgeneral position, generalizing earlier work to higher dimensions and more complex hypersurface configurations.

Findings

Established upper bounds for the number of hypersurfaces intersected by the Gauss map.

Extended previous results to the setting of projective subvarieties and hypersurfaces in subgeneral position.

Applied the results to problems of Gauss map uniqueness and shared hypersurfaces.

Abstract

In this paper, we establish some modified defect relations for the Gauss map $g$ of a complete minimal surface $S\subset\mathbb R^m$ into a $k$-dimension projective subvariety $V\subset\mathbb P^n(\mathbb C)\ (n=m-1)$ with hypersurfaces $Q_1,\ldots,Q_q$ of $\mathbb P^n(\mathbb C)$ in $N$-subgeneral position with respect to $V\ (N\ge k)$. In particular, we give the upper bound for the number $q$ if the image $g(S)$ intersects each hypersurfaces $Q_1,\ldots,Q_q$ a finite number of times and $g$ is nondegenerate over $I_d(V)$, where $d=lcm(\deg Q_1,\ldots,\deg Q_q)$, i.e., the image of $g$ is not contained in any hypersurface $Q$ of degree $d$ with $V\not\subset Q$. Our results extend and generalize the previous results for the case of the Gauss map and hyperplanes in a projective space. The results and the method of this paper have been applied by some authors to study the unicity problem of the Gauss maps sharing families of hypersurfaces.